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Theorem tgbtwntriv2 28006
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgbtwntriv2.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgbtwntriv2.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
Assertion
Ref Expression
tgbtwntriv2 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐡))

Proof of Theorem tgbtwntriv2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simprl 768 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 ∈ (𝐴𝐼π‘₯))
2 tkgeom.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
3 tkgeom.d . . . . . 6 βˆ’ = (distβ€˜πΊ)
4 tkgeom.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
5 tkgeom.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
65ad2antrr 723 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ 𝑃)
87ad2antrr 723 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐡 ∈ 𝑃)
9 simplr 766 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ π‘₯ ∈ 𝑃)
10 simpr 484 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 27982 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐡 = π‘₯)
1211adantrl 713 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 = π‘₯)
1312oveq2d 7428 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ (𝐴𝐼𝐡) = (𝐴𝐼π‘₯))
141, 13eleqtrrd 2835 . 2 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 ∈ (𝐴𝐼𝐡))
15 tgbtwntriv2.1 . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 27983 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)))
1714, 16r19.29a 3161 1 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgc 27967  df-trkgcb 27969  df-trkg 27972
This theorem is referenced by:  tgbtwncom  28007  tgbtwntriv1  28010  tgcolg  28073  legid  28106  hlid  28128  lnhl  28134  tglinerflx2  28153  mirreu3  28173  mirconn  28197  symquadlem  28208  outpasch  28274  hlpasch  28275
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